Cover
title: Guide for the Economic Design of Circular Metal Silos
author: Rotter, J. M.
publisher: Taylor & Francis Routledge
isbn10 | asin:
print isbn13: 9780203477816
ebook isbn13: 9780585447766
language: English
subject Silos--Design and construction.
publication date: 2001
lcc: TH4935.R68 2001eb
ddc: 633.2/0868
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Guide for the Economic Design of Circular Metal Silos
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Guide for the Economic Design of Circular Metal Silos
J.Michael Rotter
London and New York
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First published 2001 by Spon Press
11 New Fetter Lane, London EC4P 4EE
Simultaneously published in the USA and Canada by Spon Press
29 West 35th Street, New York, NY 10001
Spon Press is an imprint of the Taylor & Francis Group
This edition published in the Taylor & Francis e-Library, 2003. © 2001 Crown Copyright 2001. Published by permission of the Controller of Her Majesty’s Stationery Office.
All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers.
The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.
Disclaimer: The opinions expressed in this Guide are those of the author alone.
Every effort has been made to ensure accuracy in this Guide, but the publisher, the British Materials Handling Board and the author cannot accept responsibility for any loss, damage or other consequence resulting from the use of this information. Anyone making use of the information or material contained in this Guide, in whole or in part, does so at his or her own risk and assumes any and all liability from such use.
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging in Publication Data
Rotter, J.M. (J.Michael)
Guide for the economic design of circular metal silos/J.M.Rotter. p. cm.
Includes bibliographical references.
1. Silos—Design and construction. I. Title. TH4935 R68 2001
633.2'0868–dc21 00–059512
ISBN 0-203-47781-2 Master e-book ISBN ISBN 0-203-23547-9 (OEB Format) ISBN 0-419-23460-8 (alk. paper)
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Contents
List of figures ix
List of tables xiii
Disclaimer xiv
Foreword xv
Acknowledgements xvii
1 Introduction 1
1.1 Range of uses of silos 1
1.2 How to use this Guide 1
1.3 Scope 2
1.4 Underlying knowledge of silos needed for good structural design 4
1.5 Key problems for the design 6
1.6 The philosophy of structural design and its implications 10
1.7 Standards and reference documents 11
1.8 Structural reliability classes of silos in Eurocode 3 Part 4.1 13
1.9 Design expertise of the designer 14
1.10 Explanatory notes on terminology and notation 14
2 Definitions and notation 22
2.1 Definitions 22
2.2 Notation 34
3 Information on silo structure and solids required for design 39 3.1 Information to be supplied by the client 39 3.2 Key design decisions to be made by the designer 40
3.3 Summary documentation 42
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4 Data on solids required for the design 43
4.1 Loads on the structure and bulk solids material properties 43
4.2 Table of material properties 44
4.3 Variability of material properties 44
4.4 Bulk unit weight γ or density and its variation 46
4.5 Wall friction coefficient, μ 47
4.6 Lateral pressure ratio λ, K or k 48
4.7 Effective angle of internal friction, ϕi 51
4.8 Cohesion: flow function 52
4.9 Angle of repose, ϕi 56
4.10 Particle size distribution 57
4.11 Dilation characteristics 57
4.12 Maximum flow pressure multiplier Co 57
4.13 Solids trajectory on filling and eccentricity of filling ei 59
5 Guaranteeing flow and trouble-free operation: arching, ratholing and dead zones 61
5.1 General 61
5.2 Solids property characterisation 61
5.3 Ratholing 61
5.4 Arching across the outlet 64
5.5 Self-cleaning silos: dead zones 73
5.6 Flow rate: maximum achievable 74
5.7 Noises and vibration from silos: honking, banging and quaking 75
6 Flow modes and flow channel geometries 76
6.1 Definitions of flow modes 76
6.2 Advantages and disadvantages of different flow modes 78
6.3 Assumptions of the assessment 79
6.4 Criteria for mass flow 80
6.5 Criteria for pipe flow and mixed flow: concentric channels 80 6.6 Eccentric flow channels: assessment of geometry 85
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7 Wall pressures on filling and storing 89
7.1 Assumptions of the assessment 89
7.2 Division of silo geometries 93
7.3 Filling pressures on vertical walls of silos 93 7.4 Pressures on flat bottoms after filling 102
7.5 Filling pressures on hoppers 102
8 Wall pressures during concentric discharge 108 8.1 Background to concentric discharge pressures 108 8.2 Discharge pressures on vertical walls 109 8.3 Discharge pressures on conical hoppers 113
9 Wall pressures under eccentric pipe flow discharge 117
9.1 Introduction 117
9.2 Assumptions of the assessment 117
9.3 Assessment of flow channel plan geometry 118 9.4 Pressures within the flowing channel 120 9.5 Pressures on the walls exerted by the static material 121 9.6 Eccentric discharge pressures in hoppers 122
10 Structural analysis of the silo 123
10.1 Introduction 123
10.2 Types of analysis 125
10.3 Load combinations and partial factors 127
10.4 Cylindrical walls 128
10.5 Conical hoppers 135
10.6 Rings and transition junction with uniform support 141
10.7 Silos on discrete supports 148
10.8 Ring girder analysis 151
11 Strength assessment of the structure 156
11.1 Introduction 156
11.2 Cylindrical walls 158
11.3 Conical hoppers 174
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12 Symptoms of classic structural failure modes 184
12.1 Introduction 184
12.2 Actions following a failure 184
12.3 Failure due to excessive axial compression 185 12.4 Failure due to eccentric discharge 187 12.5 Failure due to excessive wind loading 187 12.6 Failure due to development of internal vacuum 187
12.7 Failures in hoppers 188
12.8 Failures in rings and supports 188
Appendix A Specification data sheet for a silo 189
Appendix B Properties of solids for operational decisions 191
Appendix C Tests for flow properties 193
Appendix D Properties of solids for loading calculations 204
Appendix E Internal explosion pressure relief 213
Appendix F Simple outline structural design for silos in Reliability Class 1 214
Appendix G Physical properties of bulk solids 221
References 222
Author index 229
Subject index 231
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List of figures
1.1 The role of bulk solids storage in industrial processes 2
1.2 Silo terminology 3
1.3 Flow patterns (after Eurocode 1 Part 4) 5
1.4 Forms of symmetric flow patterns (as used in this Guide) 5 1.5 Bounds between mass flow and funnel flow hoppers (Eurocode 1 Part 4) 6
1.6 Causes of loss of function in silos 7
1.7 Systematic causes of asymmetry in pressures 8
1.8 Flow obstructions 9
1.9 Elemental geometry of a silo 19
4.1 Upper and lower characteristic values of a material property 43 4.2 Jenike shear cell for measurement of wall friction and internal friction 49 4.3 Global slice equilibrium of solids in a vertical walled silo 50 4.4 Direct measurement of lateral pressure ratio in oedometer or lamdameter 50 4.5 Estimate of lateral pressure ratio λ from internal friction angle ϕi 51 4.6 Change in shear force to cause failure with increasing normal force 53
4.7 Frictional and cohesive solids in shear 53
4.8 Development of cohesion in a granular solid 54
4.9 Effective angle of internal friction 54
4.10 Flow function FF for a solid 55
4.11 Edinburgh Cohesion Tester for direct measurement of the unconfined compression strength 55 4.12 Effect of moisture content on cohesion of a typical coal 56 4.13 The angle of repose of a bulk solid used in this Guide 56
4.14 Particle size distribution 57
4.15 Flow patterns: densely and loosely packed solids 58 4.16 Trajectory of solids flow into silo and filling eccentricity 60
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5.1 A rathole 62
5.2 Definition of the hopper notation and outlet coordinate x0 62 5.3 Sample flow function FF for a fine particulate solid 63 5.4 Stresses in silo for analysis of cohesive arching 65
5.5a Hopper flow factor: ϕi=30° 67
5.5b Hopper flow factor: ϕi=35° 67
5.5c Hopper flow factor: ϕi=40° 68
5.5d Hopper flow factor: ϕi=45° 68
5.5e Hopper flow factor: ϕi=50° 69
5.5f Hopper flow factor: ϕi=55° 69
5.5g Hopper flow factor: ϕi=60° 70
5.5h Hopper flow factor: ϕi=65° 70
5.6 Bulk solid flow function FF compared with hopper flow factor ff 71 5.7 Finding the critical cohesive strength σcc 71
6.1 Eurocode descriptions of flow patterns 76
6.2 The conditions for mass flow and funnel flow 80 6.3 Concentric flow patterns: mass flow, pipe flow and mixed flow 82 6.4 Types of pipe flow channel: concentric and eccentric 83 6.5 Types of mixed flow channel: concentric and eccentric 83 6.6 Characteristic form of assumed flow channel geometry 84
6.7 Some causes of eccentric flow channels 85
6.8 Characteristic form of eccentric flow channel 86
6.9 Geometry of an expanded flow silo 88
7.1 Geometry definitions used in Eurocode 1 Part 4 1995 90 7.2 Distribution of normal wall pressure in barrel after filling 94 7.3 Pressures in symmetrically filled squat silos 96
7.4 Eccentricities of filling and discharge 98
7.5 Unsymmetrical local patch pressure 98
7.6 Initial pressure modification for blending and rapid filling 101 7.7 Filling pressures on steep and shallow conical hopper walls 104 7.8 Pressure increase factors for dumping into hoppers 106
8.1 Aspect ratio of a silo 111
8.2 Flow pressure multiplier or load magnifier for silos of different aspect ratios 112 8.3 Normal pressures on the walls of a conical hopper during symmetrical mass flow discharge 114 8.4 Limiting flow channel geometry for use of hopper filling pressures during discharge 115 9.1 Eccentric pipe flow discharge with stylised pressure pattern 118
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9.3 Design pressure distribution around silo circumference at any level 121
9.4 Eccentric flow channel in a hopper 122
10.1 Membrane and bending stress resultants in cylindrical wall 124
10.2 Wind pressures on circular silos 133
10.3 Symmetrical loading membrane stress resultants in conical hopper 136 10.4 Vertical equilibrium of conical hopper 138
10.5 Hopper/cylinder transition junction 143
10.6 Circumferential membrane stresses near junction 143 10.7 Importance of alignment of ring at transition 144 10.8 Notation for ring compression calculation 144 10.9 Effective section of the transition junction 145
10.10 Ring compression evaluation 147
10.11 Alternative support arrangements for discretely supported silos 149 10.12 Example ring forms for elevated silos 149 10.13 Traditional structural model for the barrel wall and transition ring 150 10.14 Ring girder notation and eccentricities 152
10.15 Loading applied to ring girder 153
11.1 Yield interaction between meridional and circumferential membrane stress resultants 159 11.2 Tolerances on dent imperfections for different qualities of fabrication 161 11.3 Measurement of depths Δwo of initial dimples (dent imperfections) 162 11.4 Reduction in elastic buckling strength with characteristic imperfection amplitude 164 11.5 Effect of internal pressure on buckling strength of cylinders 165 11.6 Elastic buckling strength gain with internal pressure 166 11.7 Reduction in plastic buckling strength with high internal pressure 167 11.8 Representation of local axial stress distribution around the circumference 169 11.9 Hopper transition joint: potential for rupture 176
11.10 Plastic collapse of conical hopper 177
11.11 Plastic collapse mode of transition junction 179 C1 Mean and standard deviation of the ‘‘normal” sample series 200 C2 Variation of bulk density with consolidation: typical plot 200 C3 Wall friction measured using a shear cell 200 C4 Determination of bulk solid yield loci 201/202
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C5 Flow function: typical plot 203
C6 Upper characteristic value using “normal” sample standard deviation with mean from “biased” samples 203
D1 Wall friction measurement 206
D2 Cell for the determination of bulk unit weight γ 207 D3 Cell for the determination of lateral pressure ratio λ 208
D4 Dimensions of profile steel sheeting 212
F1 Hopper global equilibrium 219
F2 Notation for simple transition junction 220
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List of tables
1.1 Classification of design situations 13
4.1 Physical properties of bulk solids: characteristic values 45
4.2 Definition of wall surface classes 48
7.1 Values of properties to be used for different wall loading assessments 91 7.2 Hopper pressure increase due to large dumped masses 107 10.1 Design situations and action combinations to be considered 128 10.2 Partial factors for actions for persistent and transient design situations 129 10.3 Harmonic coefficients for wind loading on circular silos and tanks 134 11.1 Partial factors on structural resistance 156 11.2 Nominal values of yield strength and ultimate tensile strength of steels 157 11.3 Fabrication quality classes for cylindrical shells 160 11.4 Values for the dimple tolerance parameter Uo,max 163 11.5 Values of external pressure buckling parameter Cb 171
B1 Features of granular bulk solids 191
B2 Individual granular bulk solids and their features 192 D1 Typical values of the coefficient of variation of material properties 210
D2 Definitions of wall roughness classes 211
Appendix G— Table 4.1—duplicated for easy reference 221
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Foreword
The purpose of this Guide is to assist the designers, manufacturers and owners of solids storage facilities to design and assess their silos using relatively simple rules that are compatible with the latest developments in regulatory standards. It provides explanations and advice where the standards may not do so and it should thus be a useful aid when designers use these standards. In addition, it addresses several important problems which lie outside the scope of other design guidance but which recent research provides the knowledge base to tackle with confidence.
The aim of the Guide is to improve current practice to achieve more efficient circular metal silo structures. Because the majority of competitive silo structures are circular in plan, this Guide is restricted to circular silos. This Guide is, in some senses, a successor to the Draft Design Code produced by the British Materials Handling Board (BMHB) and published in 1987 by BSI in association with the BMHB. It therefore addresses British conditions for silo construction and aims to assist the improvement of British practice in silo design and assessment.
The period in which this Guide is drafted is one of great change and formalisation of the rules for the design of silos. Whilst many national standards on silo pressures have been in existence for over a decade, no standard concerned with the structural design of metal silos of significant size has existed in any country. The national standards on silo pressures have been remarkable by the diversity of their provisions, both in terms of load magnitudes and in philosophy of design.
The advent of the European Standards for structural design of silos therefore changes the picture greatly. The European Standard on silo pressures was developed as a draft for comment in the years up to 1994, and is currently about to be transformed from a draft (ENV) into a normative (EN) document. The European Standard on the structural design of steel silos has recently been published as a standard for trial usage (ENV) over the next five years. This Guide follows the provisions of these European standards in almost all respects, so that users of the Guide may have confidence that their designs will meet their provisions.
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The Guide therefore now incorporates many of the provisions established within these draft European
documents, but retains additional helpful material from other sources and occasionally uses a better or simpler description of some phenomena, where the change makes a useful improvement to the economy, safety or simplicity of the design. In addition, this Guide addresses a much broader scope than the standards, and provides extensive background information to give some understanding of the basis of these provisions.
Many other standards, guides, texts and technical papers have been exploited in the development of this Guide. In particular, a strong relationship with the draft BMHB/BSI standard of 1987, as well as a certain debt to the Australian Standard AS3774, will be recognised by those who are familiar with the provisions of these documents.
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Acknowledgements
The author would like to express his thanks to many who have contributed much to his understanding of
particulate solid behaviour in silos, the behaviour and design requirements for shell structures, and the complex interactions between the two.
For encouragement and resources to conduct a major experimental project which has informed this Guide, the following are gratefully acknowledged: Mr Peter Middleton and Dr Nick Rolfe, the British Materials Handling Board, the Department of Trade and Industry, the Engineering and Physical Sciences Research Council (Grants GR/G59318 and GR/K 32029), and the industrial contributors and collaborators British Steel, Tarmac
Roadstone, British Gypsum, Brunner Mond, Cerestar, Dupont, Kemutec Braby-Fuller, Mainetti, and RJB Mining.
For many valuable insights into design provisions needed for good silo design, the author thanks all his colleagues who shared the tasks of drafting the following standards and guides:
• Australian “Structural Design of Steel Bins for Bulk Solids”, Australian Institute of Steel Construction, 1983. • Australian “Guidelines for the Assessment of Loads on Bulk Solids Containers”, Institution of Engineers,
Australia, 1986.
• ‘‘Design of Steel Bins for the Storage of Bulk Solids”, University of Sydney, 1985.
• Australian Standard “Loads on Bulk Solids Containers”, AS 3774–1996, Standards Association of Australia, 1996.
• American Concrete Institute Standard “Standard Practice for Design and Construction of Concrete Silos and Stacking Tubes for Storing Granular Materials”, ACI 313–91, 1991.
• ISO standard “Basis for design of structures—Loads due to bulk materials”, ISO DIS 11697, International Standards Organisation, 1992.
• Eurocode 1 Part 4 “Actions on silos and tanks”, ENV 1991–4, CEN, Brussels, 1995.
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• Eurocode 3 Part 1.6 “Supplementary rules for the strength and stability of shell structures”, ENV 1993–1–6, CEN, Brussels, Sept 1999.
• Eurocode 3 Part 4.1 “Design of steel structures: Silos”, ENV 1993–4–1, CEN, Brussels, Sept 1999. • Eurocode 3 Part 4.2 “Design of steel structures: Tanks’’, ENV 1993–4–2, CEN, Brussels, Sept 1999. An additional group who deserve special thanks are those who collaborated with the author in the European Concerted Action in Silos Research CA-Silo in the years 1992–6. A summary of the state of knowledge from a research perspective may be read in the book produced by this group [Brown and Nielsen, 1998].
Personal thanks are expressed to those to whom the author is particularly indebted on the subject of silos and the solids they store: Dr Jörgen Nielsen, Professor Peter Arnold, Professor Alan Roberts, Professor Richard Greiner, Professor Herbert Schmidt, Professor Bob Lohnes, Professor Jin-Guang Teng, Dr Max Blackler, Dr Jin Ooi, Dr Jian-Fei Chen, Dr Zhijun Zhong and Chris Brown. The contribution of Dr Jian-Fei Chen, who made electronic versions of many of the drawings which appear in this document and numerically re-solved Jenike’s equations for hopper flow, is especially noted. Special thanks are due to Dr Jörgen Nielsen, who read the first draft most thoroughly and offered many helpful suggestions. Many others certainly deserve individual mention (notably staff at the University of Sydney), but the list would indeed be long. The author apologises to all who may recognise their own contribution to this document, but who are not individually acknowledged.
Finally, the author expresses his thanks to his colleagues at the University of Edinburgh who have borne many additional burdens for him with generosity and patience so that this document could be completed. For the good parts of the Guide, the above are gratefully thanked. For its shortcomings, the author offers humble apologies and invites proposals for its improvement.
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Chapter 1 Introduction
1.1 Range of uses of silos
The term silo is used in this Guide to refer to all vessels for the storage of large quantities of granular bulk solids. The term “silo” is thus used here for containers which may be variously referred to as bins, bunkers, hoppers or grain tanks (US usage).
Granular bulk solids are used in a wide variety of industries. Their use chiefly arises from the need to provide a buffer between one transport activity or chemical process and another (Fig. 1.1). The experience which
underlies this Guide is derived from applications in the mining, power generation, steel making, quarrying, plastics, chemical processing, food processing, whisky production, farming and agricultural industries, where materials have been stored ranging from agricultural grains and their flours to iron ore pellets, many coals, crushed rocks and mineral ores, gypsum, plastic and polyethylene pellets and powders, and chemical process powders. The recommendations should be interpreted with this breadth and these limitations in mind.
1.2 How to use this Guide
This Guide is intended to assist designers of new installations and the assessors of existing installations to make up-to-date evaluations leading to reliable economic silos.
For those who are inexperienced in silo design, it is most desirable that the entire document should be read before attempting the task: there are many pitfalls, many strange paradoxes and a number of phenomena which are counter-intuitive in silos, so the task should not be taken on casually.
For those who are already experienced in silo design, the Guide should provide useful additional advice and guidance, together with new methods of assessing silos for features not previously standardised, and explanations which assist with the use of the Eurocode standards.
Most silo designers are either well experienced in solids handling but less experienced in structural design, or vice versa. It is hoped that each group
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Figure 1.1 The role of bulk solids storage in industrial processes.
will gain something from the material presented here that deals with the other category of problem.
The layout of the document is intended to make it easier for the designer to identify the parts of the Guide which address his specific need, and it is hoped that sufficient cautionary cross-references have been introduced to ensure that it is not necessary to have read every word to gain useful assistance from the Guide.
1.3 Scope
This Guide is concerned with the flow, wall pressures and structural design assessment of industrial silos
constructed from metal (steel, stainless steel or aluminium), used for the storage of large quantities of particulate bulk solids. The typical form of such a structure is shown in Fig. 1.2, though this Guide covers a wider range of forms than is indicated. The provisions for flow channel geometries and pressures may also be applied to concrete or composite fibre silos, but any special considerations relevant to construction in these materials have been omitted from this Guide.
The size of storage vessel considered in this Guide ranges from perhaps a tonne of solid to many thousands of tonnes. Where novel or unusual problems of structural design are encountered, the designer should refer to the more detailed structural design rules given in the Eurocodes [CEN ENV1991–4, 1995; CEN ENV1993–4–1, 1999; CEN ENV1993–1–6, 1999]. Designers of corrugated and stiffened silos will find additional valuable information in Eurocode 3 Part 4.1 [CEN ENV1993–4–1, 1999].
This Guide is designed for use chiefly with silos of circular planform. This structural form generally dominates silo construction because of its efficient use of material and ease of construction. Those wishing to use the Guide on silos of rectangular planform are referred to the Eurocodes [CEN EN V1991–4, 1995; CEN
ENV1993–4–1, 1999] where conversion information is available for pressures, and structural design rules are given for rectangular structures.
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Figure 1.2 Silo terminology.
Stored granular bulk solids encompass a wide range of materials of economic interest, with particle sizes ranging from fine cohesive powders with micron size particles to lumps of 150 mm or larger. This Guide endeavours to treat all kinds of solids which are free-flowing and do not adhere to the walls of the vessel.
While this Guide considers the effect of moisture in a limited manner, it does not apply to slurries, pastes or other systems in which the voids between the particles are substantially filled with liquid. No satisfactory limit can be defined to the moisture content for which this Guide applies, because the effect of a given percentage of moisture varies very much from one solid to another (20% might be acceptable for handling a wheat, but 15% moisture might represent complete saturation in some sands). A powder with a very
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high surface area to volume ratio may be especially susceptible to small moisture contents.
This Guide addresses the key problems of structural integrity and solids flow (as set out in Section 1.5), with advice on:
(a) Silo design to ensure reliable, steady and complete discharge of solid from the vessel;
(b) Methods of estimating the forces and pressures applied to the silo walls by the solid: these correspond to worst event values during the life of the silo;
(c) Structural design of simple circular planform metal silos.
Silos are classified according to size, bottom geometry, loading complexity and aspect ratio, following the classifications used in both Eurocode 1 Part 4 [CEN ENV1991–4, 1995] and the Australian standard [AS 3774– 1996]. Materials are classified in several different ways according to their relevant properties.
More detailed procedures are required for larger silos and for very cohesive materials. Eccentric flow conditions are considered in this Guide, but the necessary structural design calculations are complicated, and should only be attempted by designers with considerable experience of silo design and the structural analysis of shells. This Guide is based on many years of research work by a large community of researchers throughout the world, but many matters remain poorly understood and continue to provoke debate amongst them. However, silos must be designed, despite the many uncertainties, so this Guide attempts to provide sound advice on matters which are generally well agreed amongst those who have the largest body of relevant experience.
This Guide has been prepared with great care, and with reference to many other documents and standards. Nevertheless, the author and the British Materials Handling Board cannot be held responsible for any adverse consequences of advice given in this Guide.
1.4 Underlying knowledge of silos needed for good structural design
When a silo is filled with solid, the wall pressures are often quite close to the values defined by Janssen’s
analysis [Janssen, 1895], subject to careful assessment of appropriate material parameters. However, the process of discharging the silo can lead to much higher pressures, or else to locally low pressures, both of which can produce much more serious conditions for the structure.
The flow of solids during discharge can be in many different patterns (Figs 1.3 and 1.4), and the flow pattern is known to affect the wall pressures
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Figure 1.3 Flow patterns (after Eurocode 1 Part 4).
Figure 1.4 Forms of symmetric flow patterns (as used in this Guide).
markedly. However, the flow pattern is not yet easily predicted with certainty, except for the limiting cases of short squat geometries (pipe flows), and silos with steep smooth hoppers (mass flow).
In concrete silos, which are relatively thick walled but weakly reinforced, it is the pressure normal to the wall which dominates the design, and wall bending is spread quite widely. Metal silos generally have much thinner walls, and most failures are caused by buckling. The compressive stresses
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that lead to buckling failures arise firstly from solids friction against the wall, but also from unsymmetrical pressures during discharge, and from wind pressures when empty or accidental partial vacuum. The buckling strength is usually very sensitive to fabrication imperfections, which often cannot be known with certainty at the design stage. Naturally, a silo built in a factory environment using an established repeated procedure has
different imperfections from those in a one-off site constructed structure, and this difference affects the strength and thus the reliability of the construction.
A circular silo is a shell structure, so unsymmetrical pressures not only induce local bending, but affect the whole silo with membrane stresses [Rotter, 1999]. Unfortunately, most rules in silo loading standards are
currently concerned predominantly with symmetrical pressure loadings and pay little attention to unsymmetrical pressure conditions. The rules of Eurocode 1, to which this Guide refers, therefore defines some unsymmetrical pressures, but these are very simplified. In addition, the measured patterns of unsymmetrical pressures in full-scale silos [Harden et al., 1984; Schmidt and Stiglat, 1987; Ooi et al., 1990, Ooi and Rotter 1991; Rotter et al., 1995; Chen, 1996] have not yet been studied sufficiently extensively to determine their characteristics. Furthermore, the stress analysis of the silo wall requires a shell bending analysis [Trahair et al., 1983; Rotter, 1987b] which many designers are unwilling to perform, and the criteria of buckling failure have not been scientifically established for many of the resulting stress states [Rotter, 1996b].
A substantial body of new research work is needed to resolve these questions, so this Guide simply represents the current state of knowledge.
1.5 Key problems for the design
Four key problems regularly arise in the design and functioning of silos:
(a) the structural integrity of the silo may be jeopardised by pressures which are too high, too low or too unsymmetrical;
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(b) the stored solids may hang up in the container, or flow irregularly; (c) the filling or flow pattern may cause segregation of the solids; and
(d) the discharge may lead to unacceptable noises or motions (shaking, quaking and honking).
All four problems (Fig. 1.6) are related to the geometry of the silo, the material properties of the stored solids, the manner in which the solids are placed in the silo, the flow patterns which develop during discharge of solids, and any other factors affecting the stress history or dilation of the solid. However, the four phenomena have different characteristics.
The structural integrity in circular silos is often assumed to be jeopardised simply by high pressures. However, high pressures are not in themselves necessarily a serious danger to the silo structure. High pressures must be seen in the context of the structural form and the way in which the structure carries loads. Square silos and circular silos respond quite differently from
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each other, and thick-walled silos are quite different from walled silos. The remarks made here refer to thin-walled circular silos. Local unsymmetrical low pressures are usually much more damaging than very high symmetrical pressures [Rotter, 1996a and 1999]. The reasons are not easily explained, because the behaviour of shells under unsymmetrical loads is a complex subject, and because the silo is usually a multi-segment shell structure. Further extensive information on these matters may be found elsewhere [Rotter, 1985a, 1990c; Teng and Rotter, 1993; Rotter, 1996b]. If the unsymmetrical loads lead to collapse of the structure, the process which the silo feeds is arrested, and grave economic losses often ensue.
Some unsymmetrical effects are caused by quite evident features of the structural and handling arrangements (Fig. 1.7). Special note should be taken of any conditions leading to asymmetry in the structure or operating conditions (Fig. 6.7).
The stored solids may hang up in the container if one of the following three events occur: arching over the hopper outlet, ratholing or incomplete cleanout (Fig. 1.8). These are the three principal causes of arrested solids flow. Irregular flow is generally associated with one of these conditions being approached. Arching over the outlet is closely related to the development of cohesion in the solid. Ratholing is the formation of a stable hole down the entire height of a silo, leading to major loss of effective storage capacity, and consequent serious economic losses. It is also caused by cohesion in the solid. Both arching and ratholing are essentially static phenomena, so dynamic analyses are not necessary nor perhaps helpful in evaluating the causes and remedies for these problems.
Segregation caused by flow is largely addressed [Bates, 1997] by adopting a mass flow silo geometry and using careful filling procedures whenever segregation must be avoided. The criteria for evaluating when mass flow will occur were developed by Jenike and others in the 1960s [Jenike, 1961, 1964]
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Figure 1.8 Flow obstructions.
and, whilst some improvements have been made more recently [Drescher, 1992], these criteria for mass flow are now widely accepted (Fig. 1.5). In mass flow silos the flow pattern is well defined, and the conditions for mass flow are easily achieved if a hopper of adequate steepness and wall smoothness is adopted.
Funnel flow silos generally lead to segregation of the stored solid, so they are only useful for solids where this is not a problem. The flow pattern in a funnel flow silo remains a topic of serious research because asymmetries in the flow pattern can endanger the structural integrity [Rotter, 1999], but the complex form of the pressure
distribution, including both low and high pressures, must be defined before design guidance of a final nature can be given.
Shaking, quaking and honking are less well understood than the other phenomena noted above, and it is likely that several different causes lead to these effects. Silo quaking involves substantial low frequency vibration, transmitted to the surrounding building structures and causing considerable alarm. Honking is an intermittent (often unpredictable) loud report coming from the silo suddenly and unexpectedly during discharge of solids. The relationship between material properties, filling process, flow patterns, wall pressures, structural stresses and structural failure is a complex one (see
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Fig. 1.6). The mechanical characteristics of the stored solids and the process by which they are filled into the silo determine the distribution of densities and particle orientations in the silo, which strongly affect the flow pattern [Nielsen, 1983, 1998]. The filling process and flow pattern are largely responsible for segregation of the solids. The solids packing and material characteristics also determine whether arching will occur across the outlet or ratholing down the heart of the silo (see Fig. 1.8). The flow pattern in turn strongly influences the pressures on the silo wall during discharge, so these cannot be predicted with certainty unless the flow pattern itself can be predicted.
The pressures on the silo wall are, in general, unsymmetrical and nonuniform. The relationship between these pressures and the stresses which develop in the silo walls depend on shell bending phenomena, which can be very complex, and not easily understood, though they are predictable with finite element analyses. Finally, the values of silo wall stresses which will induce structural failure are only currently known for simple cases, and much research is still needed for other conditions. These failure strengths can be very dependent on both
trivially small imperfections in the wall geometry and the effect of bulk solids stiffness. They are sometimes still difficult to predict accurately even with the best modern non-linear finite element analyses.
In this rather uncertain context, this Guide is based on the principles of offering the best current design advice, and compatibility with the currently developing Eurocodes on silo loads and silo structural design.
1.6 The philosophy of structural design and its implications
Safe structural design is based on the concept that the structure must survive the most damaging events that can occur in its lifetime, with a margin of safety to allow for unknowns and uncertainties in the evaluation. Modern structural design is based on a probabilistic appreciation of both the worst events and the strength of the
structure to resist them, and is known as Limit State Design, with each extreme event leading to a limit state for the structure. In general, some limit states relate to normal operation and are termed Serviceability Limit States, whilst others relate to loss of structural integrity and are termed Ultimate Limit States. For silos, there are relatively few serviceability requirements, and most of the evaluations given here relate to ultimate limit states. To evaluate the adequacy of the structure to a given limit state, the extreme loading or set of actions applied to the structure are first determined. These extreme loads are termed ‘‘characteristic” values, and notionally have a probability of 5% of being exceeded during the life of the structure. Similarly, the strength or resistance of the structure to survive the loading without collapse is also evaluated as a characteristic value, with a notional probability of 5% that the structure will be less strong than this. These two
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characteristic values are then separated by two partial (safety) factors γ, both greater than unity: γF as a multiplier on the loads, and γM as a divisor on the resistances. The result is a “design” value of the effects of the loads on the structure, which must be less than the “design” value of the resistance the structure can provide against them (see also Section 1.10.8).
The loads applied by granular solids to silo walls depend on the properties of the solids, so the characteristic values of the loads depend on determining appropriate extreme values for these properties during the life of the structure. However, most of these properties are related to the pressures in a nonlinear manner, they combine together but are not necessarily correlated, and they influence pressures differently in silos of different
geometry. The property values defined in the processes of this Guide are therefore termed “characteristic”, but they generally do not have a probability of occurrence of 5%. Instead, the procedures used in the standards are intended to produce overall loadings with an appropriate probability of occurrence. Much research is needed to calibrate these procedures for both safety and economy.
The characteristic loads or actions on the structure are defined in Chapters 7, 8 and 9. The structural analysis, that determines the probable effect of the loads on the structure including the partial factor on loads γF, is set out in Chapter 10. The resistances of the structure to different stress states within it, together with the Limit State Verifications to ensure that the resistance exceeds the effect of the loads by an appropriate margin, are set out in Chapter 11.
1.7 Standards and reference documents
Whilst this Guide can give much useful advice, it cannot deal with all the matters which require expert
assessment. For this reason, reference is made to the following other documents as vital sources for additional information:
Eurocode on silo loads
Eurocode 1: Basis of design and actions on structures: Part 4: Actions in silos and tanks, CEN ENV 1991– 4:1995
Eurocode on steel silo structures
Eurocode 3: Design of steel structures: Part 4.1: Steel silos, CEN ENV 1993–4–1:1999 Eurocode on shell strength and stability
Eurocode 3: Design of steel structures: General rules: Part 1.6: Supplementary rules for the strength and.stability of shell structures: CEN ENV 1993–1– 6:1999
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Eurocode on steel tank structures
Eurocode 3: Design of steel structures: Part 4.2: Steel tanks, CEN ENV 1993–4–2:1999 Eurocode on general steel structures
Eurocode 3: Design of steel structures: General rules: Part 1.1: General rules and Rules for buildings: CEN ENV 1993–1–1
Eurocode on basis of design
Eurocode 1: Basis of design and actions on structures: Part 1: Basis of design, CEN ENV 1991–1 Eurocode on gravity loads
Eurocode 1: Basis of design and actions on structures: Part 2.1: Densities, self weight and imposed loads, CEN ENV 1991–2–1
Eurocode on wind loads
Eurocode 1: Basis of design and actions on structures: Part 2.4: Wind loads, CEN ENV 1991–2–4 Eurocode on structures under earthquake
Eurocode 8: Earthquake resistance design of structures: Part 4: Steel silos, tanks and pipelines, CEN ENV 1998– 4:
Australian standard on silo loads
AS 3774–1996 (1996) “Loads on Bulk Solids Containers”, Australian Standard, Standards Association of Australia, Sydney.
ACI standard on silo loads
ACI 313 (1991) “Standard Practice for Design and Construction of Concrete Silos and Stacking Tubes for Storing Granular Materials’’, ACI 313–91, with Commentary (ACI 313R–91) American Concrete Institute, Detroit.
DIN standard on silo loads
DIN 1055 (1987) “Design Loads for Buildings: Loads in Silo Bins”, DIN 1055 Part 6, Deutsches Institut für Normung, Berlin.
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ISO standard on silo loads
ISO DIS 11697 (1992) “Basis for Design of Structures-Loads Due to Bulk Materials”, ISO Draft international standard, ISO.
ISO standard on coal properties
ISO/WD 15117 (1996) “Coal: Bin Flow Properties”, ISO/TC27/SC1 N318E Working document 96/123150. BMHB guide to explosions in particulate solids
BMHB (1999).
BMHB guide to segregation
Bates, L. (1997) ‘‘User Guide to Segregation”, British Materials Handling Board, Ascot. CA-Silo research summary
Brown, C.J. and Nielsen, J. (eds) (1998) “Silos: Fundamentals of Theory, Behaviour and Design”, E & FN Spon, London.
1.8 Structural reliability classes of silos in Eurocode 3 Part 4.1
Steel silos of different sizes and complexity present very different challenges to the designer. They are therefore separated into different reliability classes in Eurocode 3 Part 4.1 [CEN ENV1993–4–1], with the following definitions:
Table 1.1 Classification of design situations
Reliability Class
Description Reliability
Class 3 *
Ground supported silos or silos supported on a complete skirt extending to the ground with capacity in excess of 5000† tonnes Discretely supported silos with capacity in excess of 1000† tonnes Silos with capacity in excess of 200† tonnes in which any of the following design situations occur:
(a) eccentric discharge (b) patch loading
(c) unsymmetrical filling Reliability
Class 2
All silos covered by Eurocode 3 Part 4.1 and not placed in another class Reliability
Class 1
Silos with capacity between 10 tonnes‡ and 100† tonnes
* Only limited advice is given in this Guide on the design of silos in Reliability Class 3. † The values given here may be changed by National Application Documents.
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The choice of reliability class will be specified by the Client, or the Relevant Authority, on the advice of the engineer, as appropriate. This Guide is intended to provide guidance on simpler designs and, where it is applied to silos in Reliability Class 3, the strict provisions of Eurocode 3 Part 4.1 should be followed.
It should be noted that Eurocode 3 Part 4.1 is limited to silos in excess of 10 tonnes capacity so that its regulatory authority is not imposed on very small silos. The guidance given here is not so restricted, but for these small silos, requirements such as robustness during transportation, fixing of attachments, etc. tend to control the detailed dimensions of the structure, rather than strength in resisting loads due to stored solids.
It should be noted that the risks associated with design situations are not differentiated according to these reliability classes, but dealt with instead by the choice of partial factor on actions (see Section 10.3).
1.9 Design expertise of the designer
The silo design should be conducted or checked in detail by a Chartered Engineer with appropriate qualifications as follows:
(a) Reliability Class 1
A Chartered Engineer with normal structural design experience for silos below 100 tonnes and using only concentric discharge.
(b) Reliability Class 2
A Chartered Engineer with specialist silo experience for these larger silos with concentric or eccentric discharge.
(c) Reliability Class 3
A Chartered Engineer with specialist silo experience is needed for these very large silos, and careful attention should be paid to the detailed requirements of Eurocode 3 Part 4.1.
1.10 Explanatory notes on terminology and notation
1.10.1 Silo terminology
The term “silo” is used here for containers which may also be variously referred to as bins, bunkers, hoppers or tanks. Different industries and professions have developed different terminology for essentially similar
containers. The term “silo” or its close relative is used in many languages (English, French, German, Italian, Greek, Danish, etc.), is unambiguous and provides a natural choice. It is therefore adopted here.
Other terms are sometimes not so helpful: the term “bin”, used chiefly in the US and Australia for industrial storage of materials like coal and mineral ores, can be confused with waste disposal bins and certainly does not convey
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the idea that its design may require a scientific assessment of high quality. The term “bunker” carries a similar connotation, the comparison being with heavy military construction, quite at variance with modern efficient silo structural design. The term “hopper’’ is used widely in chemical processing for a storage container of any geometry (often small), but it is also very widely used to identify the converging part at the bottom of a silo (Fig. 1.2). As there appear to be no alternative terms for this converging section, it is argued here that it should be reserved for this purpose. The term “tank” is widely used internationally to identify liquid storages, but in US practice is also used for silos for grain storage. It seems better to separate the terminology between liquid and solids storage (which have very different design requirements), rather than on the basis of whether the contents are agricultural grains or chemicals or whether the container is tall or short.
1.10.2 Flow pattern terminology
The terminology for flow patterns is a little confused at the present time, with different authors using different terms for the same pattern or using the same term with more or less precision. As it is commonly a matter of energetic debate and misunderstanding, the following comments are intended to clarify the terminology used here. They are written in the context of the figure given in Eurocode 1 Part 4 (see Fig. 1.3), followed by a fuller description of the terms used throughout this Guide.
“Mass flow” was clearly defined long ago as involving all the solid in a state of motion when the outlet is opened. It is helpful to reserve “mass flow” for this condition alone (see Fig. 1.4a).
“Expanded flow” silos (see Fig. 1.4d) present some terminological difficulties, as the lower hopper must be designed as a “mass flow hopper”, but the higher part involves stationary material. It is better not to refer to this container as a mass flow silo. The ASAE standard helpfully distinguishes between terms for hopper flow types and terms for silo flow patterns.
The term “funnel flow” was widely used by Jenike and later authors [e.g. Jenike, 1961; Jenike et al., 1973; Arnold et al., 1980; Roberts, 1998] for all flow patterns which are not mass flow. The need for a distinction between internal flows (see Fig. 1.4c) and flows with an effective transition against the silo wall (see Fig. 1.4b) was less clear in the 1970s than it is now, and because it has proved difficult to find a reliable criterion that can predict whether an effective transition will occur, authors have naturally been discouraged from making the distinction.
The French standard [AFNOR P22–630, 1992] and the ASAE standard [ASAE, 1989] restrict the term “funnel flow” to patterns with no effective transition. The DIN, ISO, Australian and Eurocode standards have adopted the Jenike view that all non-mass flows are funnel flows, and this accords with the widely used Jenike diagram (see Fig. 1.5) which indicates hopper
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slopes and wall frictional conditions required to distinguish mass flow from other flows, which are termed “funnel flow”. It therefore seems right to use “funnel flow” to mean all flows which are not mass flow, in line with Eurocode 1 [ENV 1991–4, 1995].
The term “core flow” is used in the BMHB draft code [BMHB, 1987] and Safarian and Harris [1985] and the German standard [DIN1055, 1987] “Kernfluβ’’ with the same meaning as Jenike’s wider use of “funnel flow”, though Nedderman [1992] restricts “core flow” to flows involving an effective transition. The term “funnel flow” is adopted here because of its apparently wider acceptance, but with an equivalence to “core flow” to match the previous BMHB advice.
A distinction is, however, needed between steep sided flow channels without an effective transition (see Fig. 1.4c) and wider channels which intersect the wall causing an effective transition (see Fig. 1.4b).
Flows that involve steep channels are termed “internal flow” by the draft Eurocode 1 and ISO standards. These standards do not cover highly eccentric outlets, so they do not consider steep sided flows which are in contact with the wall and are therefore not internal (a common feature of eccentric discharge). Thus, “internal flow” is misleading for these eccentric channel geometries, and a different term is desirable. Here, the term “internal flow” is reserved for truly internal flows. The French and ASAE standards use the term “funnel flow” for an internal flow, which must be rejected for the reasons given above. Other candidate terms are “pipe flow” and “chimney flow” (by adaptation of the French “cheminée” meaning a ship’s funnel). The term “pipe flow” is adopted here, though its apparent connection to the term “piping” used by Jenike [1961] and in the BMHB draft code [1987] to mean a stable empty channel or rathole should be avoided.
Where the pipe flow is internal, the flow has little effect on the wall pressures, and is here termed an “internal pipe flow”. Where it is in contact with the wall, it has a serious effect in locally reducing pressures and is here termed “eccentric pipe flow” (see Fig. 6.4) (Chapters 6 and 9). The “pipe” flows are separated into the
categories of “parallel pipe flow” and “taper pipe flow” to identify uniquely those common cases in which the channel has sides which are very close to vertical for the greater part of the silo (see Figs 6.3–6.5).
The condition of flow involving an effective transition (Fig. 1.4b) is variously known as “plug flow” [ASAE, 1989] “semi-mass flow” [Munch Andersen and Nielsen, 1986], “core flow” [Nedderman, 1992] or “mixed flow” [AFNOR P22–630, 1992]. The term “semi-mass flow” arises from the substantial segment of mass flowing material in the upper part of the silo, but it leads to confusion because of the reference to mass flow, so it is not used here. The term “core flow” usually has a wider meaning (see above), so it is inappropriate. The term “plug flow” is used by Drescher [1991] to mean
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an internal pipe flow with steep sides, which adds to the confusion. It probably originally referred only to the upper region of the flow zone in which mass flow is occurring (the translation of plug into French “tampon” provides a little challenge of explanation). The simpler term “mixed flow” appears to give a clearer direct meaning to a mixture of mass flow at the top and pipe flow at the bottom and so it is adopted here, and is recommended as a clear unique descriptor for flow patterns with an effective transition.
The terms adopted here are defined in Fig. 1.4. A more detailed description is given in Chapter 6 (see Figs 6.3– 6.5).
1.10.3 Stable empty channel terminology
The term “rathole” has been widely used to describe an empty channel above an outlet. The same phenomenon was termed “piping’’ by Jenike [1961, 1964] and the BMHB draft code [1987]. The former is adopted here for its evocative lack of ambiguity, and its wider acceptance throughout the world.
1.10.4 Lateral pressure ratio
The ratio of the local wall pressure to the mean vertical stress developing in the solid in a silo was first adopted as a parameter by Janssen [1895]. It is universally known as the lateral pressure ratio, but the symbol used to represent it is widely used in speech to avoid the three-word title. Here practice differs considerably. This ratio is more important in soil mechanics, where it is widely termed the ratio K. Jenike followed this practice and adopted k, which has since been used in the ACI and Australian standards [ACI 313, 1991; AS 3774, 1996], but European practice has generally favoured the use of λ [DIN 1055, 1987; BMHB, 1987; AFNOR P22–630, 1992]. It is unfortunate that this notation is also used for two other purposes which tend to cause some conflict or ambiguity. First, λ already has an important usage as the symbol adopted world-wide throughout the huge field of structural engineering to characterise the normalised slenderness, not just for shells but for columns, beams and other structures too (see Chapter 11). This poses a conflict for thin metal silos, where buckling often governs the design. Secondly, it is becoming increasingly important to take a critical state theory view of bulk solids behaviour, and λ is a key material parameter in critical state models, where it characterises the
compressive effect of consolidation [Schofield and Wroth, 1968; Muir Wood, 1990; Atkinson, 1993]. However, despite the above argument, λ is so widely used in European practice to refer to the lateral pressure ratio that it is difficult to change. In this Guide, the notation λ has finally been adopted for the lateral pressure ratio, though with some misgivings. Although K is used in the draft Eurocode 1 Part 4 [ENV 1991–4, 1995], it seems unlikely to survive in the final standard.
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1.10.5 Flow function stresses and friction angles
The flow function for a solid is defined in relation to the major consolidating stress which the solid has
experienced and the resulting unconfined yield stress that it develops as a consequence. A natural notation for these stresses would seem to be σc for the consolidating stress and σu for the unconfined yield stress. However, Jenike chose a different notation, incompatible with the above, with fc (sometimes written σc) representing the unconfined yield strength and σ1 (corresponding to the major principal stress in any stress system and therefore a rather unhelpful notation) for the major consolidating stress. This choice of notation seems particularly unfortunate, especially as silos have many conditions in both the solid and the structure where a biaxial or triaxial stress analysis is needed and σ1 should be available as an unambiguous descriptor of the local major principal stress. As a consequence, the terminology adopted here has renamed the major consolidating stress as
σm. It may be noted that some authors have attempted to use a notation like the natural one above [e.g.
Nedderman, 1992 (see list below)], leading to multiple meanings for the notation σc.
Two friction angles are regularly needed in silo evaluations. These represent the internal friction of the solid and the friction against the wall. The first analysis of a granular solid against a wall was probably that of Coulomb [1773], in which the notation of ϕ for the internal friction angle of the solid and δ for the friction angle between the solid and wall were used. This usage and meaning has been very widespread in geotechnical engineering throughout the present century. It is unfortunate that Jenike, perhaps with good intentions, achieved an inversion of this notation, adopting δ for the effective angle of internal friction and ϕ for the wall friction angle. Because many, but not all, engineers involved with bulk solids handling had no background in geotechnical engineering, the two contradictory notations are now in widespread use. In an attempt to avoid confusion, the notation δ is not used in this Guide, but subscripts are used to identify the effective internal friction angle as ϕi, the wall friction angle as ϕw and the angle of repose of the surface as ϕr.
1.10.6 Radius of silo barrel and hydraulic radius
The ratio of the area to the perimeter of a silo is critical to the evaluation of the pressures within it. It is most unfortunate that the term “hydraulic radius” was long ago chosen to represent this ratio, since there is no shape for which this ratio is indeed a useful radius. Since circular shapes are more common than most others, it would have helped had the hydraulic radius been defined as (2A/U), which would have matched the radius of a
cylinder. However, many authors writing on silo pressures have used r to characterise the hydraulic radius of the silo (being half the cylinder radius), whilst the huge literature on shell structures most commonly also uses r to characterise
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the cylinder radius. Faced with this incompatibility, this author therefore chose to use R for the cylinder radius in his writings to avoid ambiguity. However, the Eurocode committees have chosen differently. Eurocode 1
fortunately avoids using the term hydraulic radius and writes instead (A/U), but the Eurocode 3 committees have chosen r as the real radius of cylinders in all applications (e.g. tanks, towers and chimneys). This document therefore follows this convention, but the reader is warned to take care not to suppose that r represents the hydraulic radius.
The notation for the principal dimensions of a silo, used throughout this Guide, is shown in Fig. 1.9.
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1.10.7 Coordinate systems
The vertical coordinate used in defining silo pressures has long been z in most countries in the world. Shell analysts have long used a cylindrical coordinate system (r, θ, z) to analyse structures in the form of silos, so the z axis is compatible between these two descriptions. However, in the development of Eurocode 3 for steel structures, the coordinate system for a structural member was chosen to have x as the longitudinal axis, and a shell was deemed to be a member in this sense. As a result, Eurocode 3 uses x for the vertical coordinate through the silo, whilst Eurocode 1 uses z. In this Guide, an attempt has been made to include both systems, to provide the maximum consistency with the standards. Thus the pressures are defined in terms of z, but the structural analysis evaluates stresses in the same direction as x direction stresses, and conducts the resistance checks in terms of these. As a consequence, some equations in Chapter 10 may appear a little odd.
1.10.8 Limit state terminology and notation
Because this Guide may be used by some designers who do not have a modern structural engineering training, the following short explanation is given of the limit state terminology and notation found in this Guide. Three subscripts are normally used, the first to indicate the direction of a stress, the second whether it is on the loading or resistance side of the assessment, and the third the significance of the stress in the probability distribution [ENV 1991–1, 1994].
The stresses or stress resultants (forces and moments) in the structure act in different directions, so the first subscript is used to indicate a direction. In general, this is x when in the direction of the axis of the silo, θ when in the hoop or circumferential direction, and ϕ when in the sloping direction of the meridian of the hopper. For shears, two subscripts are used in this role. Sometimes, it is necessary to combine stresses acting in different directions into a von Mises effective stress, and this is given the subscript e.
The structural evaluations undertaken in this Guide are based on the Eurocode limit state format. The second subscript identifies whether it is a load effect or resistance that is being determined. The actions (loads) on the structure induce a set of stress resultants within the structure which are given the subscript S (or for stresses E). The strengths or resistances that the structure can offer to those stress resultants are given the subscript R. The third subscript indicates the probability of this magnitude arising. The actions (loads) and resistances are both evaluated as values that have a notional 5% probability of occurring: these are termed “characteristic” values and given the subscript k. When values for checking the safety of the design are obtained, the actions have been increased by γF from the characteristic value to a “design” value, and the resistances have been decreased by
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γM from the characteristic value to a design value. Both of these design values are given the subscript d. This
subscript is also used for simpler measures of resistance, as in c for the elastic buckling strength of a perfect structure and p for the plastic collapse resistance.
The check for the structural integrity of the design is that the values S should always be less than the values R. 1.10.9 Consistent units
The equations used throughout this Guide are dimensionally consistent. This means that it is not necessary to be sure to adopt special units in any particular equation. However, a consistent set of units for calculations is desirable.
The following two alternative sets of recommended consistent units are taken from Eurocode 3 Part 4.1 [ENV 1993–4–1, 1999].
dimensions and thicknesses : m mm
unit weight : kN/m3 N/mm3
forces and loads : kN N
line forces and line loads : kN/m N/mm
pressures and area distributed actions : kPa MPa
unit mass : kg/m3 kg/mm3
acceleration : km/s2 m/s2
membrane stress resultants : kN/m N/mm
bending stress resultants : kNm/m Nmm/mm
stresses and elastic moduli : kPa MPa
Conversion factors: 1 MPa=1N/mm2 1 kPa=1kN/m2 1 bar ≈ 100 kPa
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Chapter 2
Definitions and notation 2.1 Definitions
The following definitions apply throughout the Guide. Where the term has more than one distinct meaning, the alternatives are separated as (a), (b), (c), etc. Terms which are defined elsewhere in the definitions list are shown in italics.
Aeration Where air is entrained in a fine particulate solid (powder), the solid is said to be aerated. This can arise either from deliberate air injection, from pneumatic conveying or from the filling or
discharge processes.
Angle of repose
The steepest angle that can be measured on a free surface on the top of the granular solid (defined in Section 4.9).
Apex of hopper
A hopper is terminated by the outlet. However, the geometry of the cone is defined in terms of the geometric apex, which is a point in space below the outlet (Figs 1.9, 7.7 and 8.3).
Arching Formation of an arch, bridge or dome in the stored solids over the silo outlet, usually in the hopper (Fig. 1.8) and with a void beneath it. This Guide refers only to stable static arches, which arrest the flow.
Axisymmetric shell
A shell structure whose geometry is defined by the rotation of a meridional line about a central axis (cylinder, cone, sphere, etc.).
Barrel sectionThe upper part of a silo with vertical parallel walls. Bending
moment
The bending moment per unit width of wall is a couple within the shell wall (no net force resultant). A shell carries bending moments in both the vertical and circumferential directions. The direction of the bending
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moment is defined by the direction of the stresses it induces (note the contrast with beam theory where bending is about an axis within the beam).
At any point in a shell, there are three bending moments (two direct moments and a twisting moment).
Bending stress The concept of a bending stress is useful under conditions where the silo wall is elastic. The
bending stress then varies linearly through the wall thickness from a tensile value on one face to an equal compressive value on the opposite face. Bending stresses affect conditions of yielding of the wall but do not affect the buckling response (cf. membrane stresses).
The bending stress distribution is characterised by the value of bending stress occurring on the outer surface (tensile positive on the outer surface).
Buckling The ultimate limit state for the structure, where it suddenly loses its stability under compression and/or shear. It leads either to very large displacements or to structural failure.
Bulk density The mass of a quantity of particulate solid divided by the total volume including interstitial voids. The bulk density depends on the packing structure and the stress history.
Bulk solid This term is used to refer to particulate solids when they are handled in large quantities. Whilst the definition is not strict, it is widely used when the conceptual image of the solid does not take account of the individual particles but treats them as a continuum.
Circumferential direction
The horizontal tangent to the silo wall at any point. It varies in direction around the silo, lies in the horizontal plane and is tangential to the silo wall irrespective of whether the silo is circular or rectangular in plan.
Cohesion The component of a particulate solid’s shear strength which is independent of the applied normal stress. A solid is termed one of low cohesion if the unconfined yield stress is less than 14 kPa after the solid has been precompressed to a stress of 100 kPa (Eurocode 1 Part 4).
Critical state The state in a particulate solid where large changes in deformations can occur without change to either the stress state or the total volume of the assembly of particles.
